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For the Wiener, Ornstein–Uhlenbeck, and Feller processes, we study the transition probability density functions with an absorbing boundary in the zero state. Particular attention is paid to the proportional cases and to the time-homogeneous cases, by obtaining the first-passage time densities through the zero state. A detailed study of the asymptotic average of local time in the presence of an absorbing boundary is carried out for the time-homogeneous cases. Some relationships between the transition probability density functions in the presence of an absorbing boundary in the zero state and between the first-passage time densities through zero for Wiener, Ornstein–Uhlenbeck, and Feller processes are proven. Moreover, some asymptotic results between the first-passage time densities through zero state are derived. Various numerical computations are performed to illustrate the role played by parameters. [ABSTRACT FROM AUTHOR]

We consider the first-passage time problem for the Feller-type diffusion process, having infinitesimal drift B 1 (x , t) = α (t) x + β (t) and infinitesimal variance B 2 (x , t) = 2 r (t) x , defined in the space state [ 0 , + ∞) , with α (t) ∈ R , β (t) > 0 , r (t) > 0 continuous functions. For the time-homogeneous case, some relations between the first-passage time densities of the Feller process and of the Wiener and the Ornstein–Uhlenbeck processes are discussed. The asymptotic behavior of the first-passage time density through a time-dependent boundary is analyzed for an asymptotically constant boundary and for an asymptotically periodic boundary. Furthermore, when β (t) = ξ r (t) , with ξ > 0 , we discuss the asymptotic behavior of the first-passage density and we obtain some closed-form results for special time-varying boundaries. [ABSTRACT FROM AUTHOR]

General methods to simulate probability density functions and first passage time densities are provided for time-inhomogeneous stochastic diffusion processes obtained via a composition of two Gauss–Markov processes conditioned on the same initial state. Many diffusion processes with time-dependent infinitesimal drift and infinitesimal variance are included in the considered class. For these processes, the transition probability density function is explicitly determined. Moreover, simulation procedures are applied to the diffusion processes obtained starting from Wiener and Ornstein–Uhlenbeck processes. Specific examples in which the infinitesimal moments include periodic functions are discussed. [ABSTRACT FROM AUTHOR]

Algorithms for the simulation of sample paths of Gauss-Markov processes, restricted from below by particular time-dependent reflecting boundaries, are proposed. These algorithms are used to build the histograms of first passage time density through specified boundaries and for the estimation of related moments. Particular attention is dedicated to restricted Wiener and Ornstein-Uhlenbeck processes due to their central role in the class of Gauss-Markov processes. [ABSTRACT FROM AUTHOR]